The 21-year-old floored Cope in the sixth round before claiming a unanimous points decision from the ringside judges.
Cope is the Harrow boxer's second consecutive unbeaten scalp after his two-round demolition of Mark Evans in February and Smith is targeting more success in the next 12 months.
"When I knocked the Welsh champion Mark Evans out in two rounds people starting asking questions about my ability to go 10 or 12 rounds," Smith told Sky Sports. "But I did 10 rounds tonight at a decent pace, I'm answering questions all the time and I'm ticking all the right boxes.
"I've been a pro now two years and I'm doing things the traditional way, Southern Area, now the English and I'd like to challenge for the British in the next 12 months.
"But it was a patient and mature performance tonight and I'm so happy to go down in history as having been an English champion."
Meanwhile, Frank Buglioni bounced back from the loss of his European belt to Sergei Khomitsky with a fine fourth-round stoppage victory over Sam Couzens.
A packed undercard at London's York Hall also saw wins for Bradley Skeete, Gary Corcoran, Tom Baker and Romeo Romaeo.